Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 350979, 20 pages doi:10.1155/2009/350979 Research Article A New Hybrid Algorithm for Variational Inclusions, Generalized Equilibrium Problems, and a Finite Family of Quasi-Nonexpansive Mappings Prasit Cholamjiak and Suthep Suantai Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand Correspondence should be addressed to Suthep Suantai, scmti005@chiangmai.ac.th Received 12 June 2009; Accepted 28 September 2009 Recommended by Naseer Shahzad We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept of W-mappings for a finite family of quasi-nonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces. Copyright q 2009 P. Cholamjiak and S. Suantai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let H be a real Hilbert space with inner product ·, · and inducted norm ·,andletC be a nonempty closed and convex subset of H. Then, a mapping T : C → C is said to be 1 nonexpansive if Tx − Ty≤x − y, for all x, y ∈ C; 2 quasi-nonexpansive if Tx − p≤x − p, for all x ∈ C and p ∈ FT; 3 L-Lipschitzian if there exists a constant L>0 such that Tx − Ty≤Lx − y, for all x, y ∈ C. We denoted by FT the set of fixed points of T. In 1953, Mann 1 introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H: x n1  α n x n   1 − α n  Tx n , ∀n ∈ N, 1.1 where the initial point x 0 is taken in C arbitrarily and {α n } is a sequence in 0, 1. 2 Fixed Point Theory and Applications However, we note that Mann’s iteration process 1.1 has only weak convergence, in general; for instance, see 2, 3. Many authors attempt to modify the process 1.1 so that strong convergence is guaranteed that has recently been made. Nakajo and Takahashi 4 proposed the following modification which is the so-called CQ method and proved the following strong convergence theorem for a nonexpansive mapping T in a Hilbert space H. Theorem 1.1 see 4. Let C be a nonempty closed convex subset of a Hilbert space H and let T be a nonexpansive mapping of C into itself such that FT /  ∅. Suppose that x 1  x ∈ C and {x n } is given by y n  α n x n   1 − α n  Tx n , C n   z ∈ C :   y n − z   ≤  x n − z   , Q n  { z ∈ C :  x n − z, x − x n  ≥ 0 } , x n1  P C n ∩Q n x, ∀n ∈ N, 1.2 where 0 ≤ α n ≤ a<1. Then, {x n } converges strongly to z 0  P FT x. Let ϕ : H → R ∪{∞}be a function and let F be a bifunction from C × C to R such that C ∩ dom ϕ /  ∅, where R is the set of real numbers and dom ϕ  {x ∈ H : ϕx < ∞}.The generalized equilibrium problem is to find x ∈ C such that F  x, y   ϕ  y  − ϕ  x  ≥ 0, ∀y ∈ C. 1.3 The set of solutions of 1.3 is denoted by GEPF, ϕ;seealso5–7. If ϕ : H → R ∪{∞}is replaced by a real-valued function φ : C → R, problem 1.3 reduces to the following mixed equilibrium problem introduced by Ceng and Yao 8:find x ∈ C such that F  x, y   φ  y  − φ  x  ≥ 0, ∀y ∈ C. 1.4 Let ϕxδ C x, for all x ∈ H.Hereδ C denotes the indicator function of the set C;thatis, δ C x0ifx ∈ C and δ C x∞ otherwise. Then problem 1.3 reduces to the following equilibrium problem: find x ∈ C such that F  x, y  ≥ 0, ∀y ∈ C. 1.5 The set of solutions 1.5 is denoted by EPF. Problem 1.5 includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem; see 9–12 and the reference cited therein. Recently, Tada and Takahashi 13 proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping T in a Hilbert space H and then obtain the following theorem. Fixed Point Theory and Applications 3 Theorem 1.2 see 13. Let H be a real Hilbert space, let C be a closed convex subset of H,let F : C × C → R be a bifunction, and let T : C → C be a nonexpansive mapping such that FT ∩ EPF /  ∅. For an initial point x 1  x ∈ C, let a sequence {x n } be generated by F  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0 ∀y ∈ C, y n  α n x n   1 − α n  Tu n , C n   z ∈ C :   y n − z   ≤  x n − z   , Q n  { z ∈ C :  x n − z, x n − x  ≤ 0 } , x n1  P C n ∩Q n x, ∀n ∈ N, 1.6 where 0 ≤ α n ≤ a<1 and lim inf n →∞ r n > 0. Then, {x n } converges strongly to P FT∩EPF x. Let A : H → H be a single-valued nonlinear mapping and let M : H → 2 H be a set-valued mapping. The variational inclusion is to find x ∈ H such that θ ∈ A  x   M  x  , 1.7 where θ is the zero vector in H. The set of solutions of problem 1.7 is denoted by IA, M. Recall that a mapping A : H → H is called α-inverse strongly monotone if there exists a constant α>0 such that  Ax − Ay, x − y  ≥ α   Ax − Ay   2 , ∀x, y ∈ H. 1.8 A set-valued mapping M : H → 2 H is called monotone if for all x, y ∈ H, f ∈ Mx, and g ∈ My imply x − y,f − g≥0. A monotone mapping M is maximal if its graph GM : {f,x ∈ H × H : f ∈ Mx} of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for x, f ∈ H × H, x − y, f − g≥0 for all y, g ∈ GM imply f ∈ Mx. We define the resolvent operator J M,λ associated with M and λ as follows: J M,λ  x    I  λM  −1  x  ,x∈ H, λ > 0. 1.9 It is known that the resolvent operator J M,λ is single-valued, nonexpansive, and 1- inverse strongly monotone; see 14, and that a solution of problem 1.7 is a fixed point of the operator J M,λ I − λA for all λ>0; see also 15.If0<λ<2α, it is easy to see that J M,λ I − λA is a nonexpansive mapping; consequently, IA, M is closed and convex. The equilibrium problems, generalized equilibrium problems, variational inequality problems, and variational inclusions have been intensively studied by many authors; for instance, see 8, 16–43. Motivated by Tada and Takahashi 13 and Peng et al. 7, we introduce a new approximation scheme for finding a common element of the set of fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings, the set of solutions of a generalized 4 Fixed Point Theory and Applications equilibrium problem, and the set of solutions of a variational inclusion with set-valued maximal monotone and inverse strongly monotone mappings in the framework of Hilbert spaces. 2. Preliminaries and Lemmas Let C be a closed convex subset of a real Hilbert space H with norm ·and inner product ·, ·. For each x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that x − P C x  min y∈C x − y. P C is called the metric projection of H on to C.Itisalsoknownthat for x ∈ H and z ∈ C, z  P C x is equivalent to x − z, y − z≤0 for all y ∈ C. Furthermore   y − P C x   2   x − P C x  2 ≤   x − y   2 2.1 for all x ∈ H, y ∈ C;seealso4, 44. In a real Hilbert space, we also know that   λx 1 − λy   2  λ  x  2   1 − λ    y   2 − λ  1 − λ    x − y   2 2.2 for all x, y ∈ H and λ ∈ 0, 1. Lemma 2.1 see 45. Let C be a nonempty closed convex subset of a Hilbert space H. Then for points w, x, y ∈ H and a real number a ∈ R, the set D :  z ∈ C : y − z 2 ≤x − z 2   w, z   a  is closed and convex. 2.3 For solving the generalized equilibrium problem, let us give the following assump- tions for F, ϕ, and the set C: A1 Fx, x0 for all x ∈ C; A2 F is monotone, that is, Fx, yFy, x ≤ 0 for all x, y ∈ C; A3 for each y ∈ C, x → Fx, y is weakly upper semicontinuous; A4 for each x ∈ C, y → Fx, y is convex; A5 for each x ∈ C, y → Fx, y is lower semicontinuous; B1 for each x ∈ H and r> 0, there exists a bounded subset D x ⊆ C and y x ∈ C ∩ dom ϕ such that for any z ∈ C \ D x , F  z, y x   ϕ  y x   1 r  y x − z, z − x  <ϕ  z  ; 2.4 B2 C is a bounded set. Lemma 2.2 see 7. Let C be a nonempty closed convex subset of a real Hilbert H.LetF be a bifunction from C × C to R satisfying (A1)–(A5) and let ϕ : H → R ∪{∞}be a proper lower Fixed Point Theory and Applications 5 semicontinuous and convex function such that C ∩dom ϕ /  ∅. For r>0 and x ∈ H, define a mapping S r : H → C as follows: S r  x    z ∈ C : F  z, y   ϕ  y   1 r  y − z, z − x  ≥ ϕ  z  , ∀y ∈ C  . 2.5 Assume that either (B1) or (B2) holds. Then, the following conclusions hold: 1 for each x ∈ H, S r x /  ∅; 2 S r is single-valued; 3 S r is firmly nonexpansive, that is, for any x, y ∈ H,   S r x − S r y   2 ≤  S r  x  − S r  y  ,x− y  ; 2.6 4 FS r GEPF, ϕ; 5 GEPF, ϕ is closed and convex. Lemma 2.3 see 14. Let M : H → 2 H be a maximal monotone mapping and let A : H → H be a Lipshitz continuous mapping. Then the mapping S  M  A : H → 2 H is a maximal monotone mapping. Lemma 2.4. Let C be a nonempty closed convex subset of a real Hilbert space H.LetT be a quasi- nonexpansive and L-Lipschitz mapping of C into itself. Then, FT is closed and convex. Proof. Since T is L-Lipschitz, it is easy to show that FT is closed. Let x, y ∈ FT and z  tx 1 − ty where t ∈ 0, 1.From2.2, we have  z − Tz  2  t  x − Tz  2   1 − t    y − Tz   2 − t  1 − t    x − y   2 ≤ t  x − z  2   1 − t    y − z   2 − t  1 − t    x − y   2  t  1 − t  2   x − y   2   1 − t  t 2   x − y   2 − t  1 − t    x − y   2  0, 2.7 which implies z ∈ FT; consequently, FT is convex. This completes the proof. Lemma 2.5 see 46. In a strictly convex Banach space X,if  x     y      λx   1 − λ  y   2.8 for all x, y ∈ X and λ ∈ 0, 1,thenx  y. 6 Fixed Point Theory and Applications In 1999, Atsushiba and Takahashi 47 introduced the concept of the W-mapping as follows: U 1  β 1 T 1   1 − β 1  I, U 2  β 2 T 2 U 1   1 − β 2  I, . . . U N−1  β N−1 T N−1 U N−2   1 − β N−1  I, W  U N  β N T N U N−1   1 − β N  I, 2.9 where {T i } N i1 is a finite mapping of C into itself and β i ∈ 0, 1 for all i  1, 2, ,N with  N i1 β i  1. Such a mapping W is called the W-mapping generated by T 1 ,T 2 , ,T N and β 1 ,β 2 , ,β N ;seealso48–50. Throughout this paper, we denote F :  N i1 FT i . Next, we prove some useful lemmas concerning the W-mapping. Lemma 2.6. Let C be a nonempty closed convex subset of a strictly convex Banach space X.Let{T i } N i1 be a finite family of quasi-nonexpansive and L i -Lipschitz mappings of C into itself such that F :  N i1 FT i  /  ∅ and let β 1 ,β 2 , ,β N be real numbers such that 0 <β i < 1 for all i  1, 2, ,N−1, 0 < β N ≤ 1, and  N i1 β i  1.LetW be the W-mapping generated by T 1 ,T 2 , ,T N and β 1 ,β 2 , ,β N . Then, the followings hold: i W is quasi-nonexpansive and Lipschitz; ii FW  N i1 FT i . Proof. i For each x ∈ C and z ∈ F, we observe that  T 1 x − z  ≤  x − z  . 2.10 Let k ∈{2, 3, ,N}, then  U k x − z     β k T k U k−1 x   1 − β k  x − z   ≤ β k  U k−1 x − z    1 − β k   x − z  . 2.11 Hence,  Wx − z    U N x − z  ≤ β N  U N−1 x − z    1 − β N   x − z  ≤ β N  β N−1  U N−2 x − z    1 − β N−1   x − z     1 − β N   x − z  Fixed Point Theory and Applications 7 ≤ β N  β N−1  β N−2  U N−3 x − z    1 − β N−2   x − z     1 − β N−1   x − z     1 − β N   x − z  . . . ≤ β N  β N−1  β N−2 ···  β 2  β 1  T 1 x − z    1 − β 1   x − z     1 − β 2   x − z    ···  1 − β N−2   x − z     1 − β N−1   x − z     1 − β N   x − z  ≤ β N  β N−1  β N−2 ···  β 2  β 1  x − z    1 − β 1   x − z     1 − β 2   x − z    ···  1 − β N−2   x − z     1 − β N−1   x − z     1 − β N   x − z   β N  β N−1  β N−2 ···  β 3  β 2  x − z    1 − β 2   x − z     1 − β 3   x − z    ···  1 − β N−2   x − z     1 − β N−1   x − z     1 − β N   x − z    x − z  . 2.12 This shows that W is a quasi-nonexpansive mapping. Next, we claim that W is a Lipschitz mapping. Note that T i is L i -Lipschitz for all i  1, 2, ,N. For each x, y ∈ C, we observe   U 1 x − U 1 y      β 1 T 1 x   1 − β 1  x − β 1 T 1 y −  1 − β 1  y   ≤ β 1   T 1 x − T 1 y     1 − β 1    x − y   ≤  β 1 L 1   1 − β 1    x − y   . 2.13 Let k ∈{2, 3, ,N}, then   U k x − U k y      β k T k U k−1 x   1 − β k  x − β k T k U k−1 y −  1 − β k  y   ≤ β k L k   U k−1 x − U k−1 y     1 − β k    x − y   . 2.14 Hence,   Wx − Wy   ≤ β N L N   U N−1 x − U N−1 y     1 − β N    x − y   ≤ β N L N β N−1 L N−1   U N−2 x − U N−2 y     β N L N  1 − β N−1    1 − β N    x − y   . . . 8 Fixed Point Theory and Applications ≤ β N L N β N−1 L N−1 ···β 2 L 2   U 1 x − U 1 y     β N L N β N−1 L N−1 ···β 3 L 3  1 − β 2   β N L N β N−1 L N−1 ···β 4 L 4  1 − β 3   ··· β N L N  1 − β N−1    1 − β N    x − y   ≤ β N L N β N−1 L N−1 ···β 2 L 2  β 1 L 1   1 − β 1    x − y      β N L N β N−1 L N−1 ···β 3 L 3  1 − β 2   β N L N β N−1 L N−1 ···β 4 L 4  1 − β 3   ··· β N L N  1 − β N−1    1 − β N    x − y     β N L N β N−1 L N−1 ···β 1 L 1  β N L N β N−1 L N−1 ···β 2 L 2  1 − β 1   β N L N β N−1 L N−1 ···β 3 L 3  1 − β 2   β N L N β N−1 L N−1 ···β 4 L 4  1 − β 3   ··· β N L N  1 − β N−1    1 − β N    x − y   . ≤  L N L N−1 ···L 1  L N L N−1 ···L 2  L N L N−1 ···L 3 L N L N−1 ···L 4  ··· L N L N−1  L N  1    x − y   . 2.15 Since L i > 0 for all i  1, 2, ,N,wegetthatW is a Lipschitz mapping. ii Since F ⊂ FW is trivial, it suffices to show that FW ⊂ F. To end this, let p ∈ FW and x ∗ ∈ F. Then, we have   p − x ∗      Wp − x ∗      β N  T N U N−1 p − x ∗    1 − β N  p − x ∗    ≤ β N   U N−1 p − x ∗     1 − β N    p − x ∗    β N   β N−1  T N−1 U N−2 p − x ∗    1 − β N−1  p − x ∗      1 − β N    p − x ∗   ≤ β N β N−1   U N−2 p − x ∗     1 − β N β N−1    p − x ∗    β N β N−1   β N−2  T N−2 U N−3 p − x ∗    1 − β N−2  p − x ∗      1 − β N β N−1    p − x ∗   ≤ β N β N−1 β N−2   U N−3 p − x ∗     1 − β N β N−1 β N−2    p − x ∗   . . .  β N β N−1 ···β 3   β 2  T 2 U 1 p − x ∗    1 − β 2  p − x ∗      1 − β N β N−1 ···β 3    p − x ∗   Fixed Point Theory and Applications 9 ≤ β N β N−1 ···β 2   T 2 U 1 p − x ∗     1 − β N β N−1 ···β 2    p − x ∗   ≤ β N β N−1 ···β 2   U 1 p − x ∗     1 − β N β N−1 ···β 2    p − x ∗    β N β N−1 ···β 2   β 1  T 1 p − x ∗    1 − β 1  p − x ∗      1 − β N β N−1 ···β 2    p − x ∗   ≤ β N β N−1 ···β 2 β 1   T 1 p − x ∗     1 − β N β N−1 ···β 2 β 1    p − x ∗   ≤ β N β N−1 ···β 2 β 1   p − x ∗     1 − β N β N−1 ···β 2 β 1    p − x ∗      p − x ∗   . 2.16 This shows that   p − x ∗    β N β N−1 ···β 2   β 1  T 1 p − x ∗    1 − β 1  p − x ∗      1 − β N β N−1 ···β 2    p − x ∗   , 2.17 and hence   p − x ∗      β 1  T 1 p − x ∗    1 − β 1  p − x ∗    . 2.18 Again by 2.16,weseethatp − x ∗   T 1 p − x ∗ . Hence   p − x ∗      T 1 p − x ∗      β 1  T 1 p − x ∗    1 − β 1  p − x ∗    . 2.19 Applying Lemma 2.5 to 2.19,wegetthatT 1 p  p and hence U 1 p  p. Again by 2.16, we have   p − x ∗    β N β N−1 ···β 3   β 2  T 2 U 1 p − x ∗    1 − β 2  p − x ∗      1 − β N β N−1 ···β 3    p − x ∗   , 2.20 and hence   p − x ∗      β 2  T 2 U 1 p − x ∗    1 − β 2  p − x ∗    . 2.21 From 2.16, we know that U 1 p − x ∗   T 2 U 1 p − x ∗ . Since U 1 p  p, we have   p − x ∗      T 2 p − x ∗      β 2  T 2 p − x ∗    1 − β 2  p − x ∗    . 2.22 Applying Lemma 2.5 to 2.22,wegetthatT 2 p  p and hence U 2 p  p. By proving in the same manner, we can conclude that T i p  p and U i p  p for all i  1, 2, ,N− 1. Finally, we also have   p − T N p   ≤   p − Wp      Wp − T N p      p − Wp     1 − β N    p − T N p   , 2.23 which yields that p  T N p since p ∈ FW. Hence p ∈ F :  N i1 FT i . 10 Fixed Point Theory and Applications Lemma 2.7. Let C be a nonempty closed convex subset of a Banach space X.Let{T i } N i1 be a finite family of quasi-nonexpansive and L i -Lipschitz mappings of C into itself and {β n,i } N i1 sequences in 0, 1 such that β n,i → β i as n →∞. Moreover, for every n ∈ N,letW and W n be the W-mappings generated by T 1 ,T 2 , ,T N and β 1 ,β 2 , ,β N and T 1 ,T 2 , ,T N and β n,1 ,β n,2 , ,β n,N , respectively. Then lim n →∞  W n x − Wx   0, ∀x ∈ C. 2.24 Proof. Let x ∈ C and U k and U n,k be generated by T 1 ,T 2 , ,T k and β 1 ,β 2 , ,β k and T 1 ,T 2 , ,T k and β n,1 ,β n,2 , ,β n,k , respectively. Then  U n,1 x − U 1 x      β n,1 − β 1   T 1 x − x    ≤   β n,1 − β 1    T 1 x − x  . 2.25 Let k ∈{2, 3, ,N} and M  max{T k U k−1 x  x : k  2, 3, ,N}. Then  U n,k x − U k x     β n,k T k U n,k−1 x   1 − β n,k  x − β k T k U k−1 −  1 − β k  x      β n,k T k U n,k−1 x − β n,k x − β k T k U k−1  β k x   ≤ β n,k  T k U n,k−1 x − T k U k−1 x     β n,k − β k    T k U k−1 x     β n,k − β k    x  ≤ L k  U n,k−1 x − U k−1 x     β n,k − β k   M. 2.26 It follows that  W n x − Wx    U n,N x − U N x  ≤ L N  U n,N−1 x − U N−1 x     β n,N − β N   M ≤ L N  L N−1  U n,N−2 x − U N−2 x     β n,N−1 − β N−1   M     β n,N − β N   M  L N L N−1  U n,N−2 x − U N−2 x   L N   β n,N−1 − β N−1   M    β n,N − β N   M . . . ≤ L N L N−1 ···L 3  L 2  U n,1 x − U 1 x     β n,2 − β 2   M   L N L N−1 ···L 4   β n,3 − β 3   M  ··· L N   β n,N−1 − β N−1   M    β n,N − β N   M ≤ L N L N−1 ···L 2   β n,1 − β 1    T 1 x − x   L N L N−1 ···L 3   β n,2 − β 2   M  L N L N−1 ···L 4   β n,3 − β 3   M  ··· L N   β n,N−1 − β N−1   M    β n,N − β N   M. 2.27 Since β n,i → β i as n →∞i  1, 2, ,N,weobtaintheresult. 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